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Page 1: Appendix A List of Symbols · Appendix A List of Symbols The following is a list of symbols used throughout the description of the sticky par-ticle star formation model (see chapter

Appendix AList of Symbols

The following is a list of symbols used throughout the description of the sticky par-

ticle star formation model (see chapter 3)

αc : Slope of the molecular cloud mass-radius relation. Eq 3.5

ch : Sound speed of the ambient gas phase

ε? : Fraction of a GMC converted to stars in a collapse. E51 : Energy ejected per SnII in

units of 1051ergs

Eb : Total energy in a supernova blast wave

Em : Total kinetic energy in molecular clouds of mass m in a given volume

fcl : Filling factor of cold clouds

fm(σ1, σ2) : Fraction of collisions between clouds with velocity dispersions σ1 and σ2

that lead to mergers

K(m,m′) : The kernel for aggregation of clouds of masses m and m′. Eq. 3.6

λ : Constant of proportionality relating cloud mass and destruction rate by thermal con-

duction. Eq. 3.50

ΛN : Normalised radiative cooling rate

Λnet : Net radiative cooling rate (ergs cm−3s−1)

Mc : Mass of a molecular cloud

Mref : Reference cold cloud mass. Eq 3.5

M?,min : Minimum allowed star mass

M?,max : Maximum allowed star mass

nb : Density internal to a supernova remnant in atoms / cm3 nc : Density of a molecular

cloud in atoms / cm3

nh : Density of the ambient medium in atoms / cm3

NSF : The slope of the schmidt law. Eq 3.36

n(m, t) : The number of clouds with masses between m and m+ dm

N(m, t) : The number density of clouds with masses between m and m+ dm

φ : Efficiency of destruction of cold clouds by thermal conduction

Q : Porosity of the interstellar medium. Sec. 3.2.6

173

Page 2: Appendix A List of Symbols · Appendix A List of Symbols The following is a list of symbols used throughout the description of the sticky par-ticle star formation model (see chapter

A. List of Symbols 174

rc : Radius of a molecular cloud

rref : Reference cold cloud radius. Eq 3.5

rb : The radius of a spherical blast wave

ρc : Mean density of molecular clouds contained in a volume

ρh : Mean density of ambient gas contained in a volume

ρth : Density at which ambient gas becomes thermally unstable

ρSFR : Volume density of star formation

η : Fraction of cloud velocity lost to ’cooling’ collision

Tb : Mean temperature inside of a supernova remnant

Tc : Internal temperature of cold clouds

Th : Temperature of the ambient gas phase

ub : Thermal energy per unit mass of supernova remnants

uc : Thermal energy per unit mass of the cold clouds

uh : Thermal energy per unit mass of the ambient phase

Σ : Cross section for collision between clouds. Eq. 3.7

Σcond : Efficiency of thermal conduction. Eq 3.46

vapp : Relative approach velocity of two molecular clouds

vstick : Maximum relative velocity for cloud merger

x : Slope of the stellar IMF

Page 3: Appendix A List of Symbols · Appendix A List of Symbols The following is a list of symbols used throughout the description of the sticky par-ticle star formation model (see chapter

Appendix BThe Green’s Function ofthe Finite DifferencedLaplacian

For some function φ, defined on a regular grid at points i, with grid spacing ∆ the

finite-difference approximation to the Laplacian at point i is given by

∇2φi ≈φi+1 + φi−1 − 2φi

∆2. (B.1)

We now note that for some function g(x), F(g(t)

)= G(k), where the notation F repre-

sents a Fourier transform, defined as

g(x) =

∫ ∞

∞G(k)e2πikxdk . (B.2)

k represents a frequency, we can write

∇2φ =∑

k

φ(k)ei2πk∆ + φ(k)e−i2πk∆ − 2φ(k)

∆2e2πikx , (B.3)

by using F(g(t − a)

)= e−i2πakG(k). Now noting that eiax = cos(ax) + isin(ax) we can

write

∇2φ =∑

k

φ(k)cos(2πk∆) + isin(2πk∆) + cos(−2πk∆) + isin(−2πk∆) − 2

∆2eikx , (B.4)

which, through symmetry, becomes

∇2φ =∑

k

φ(k)2cos(2πk∆) − 2

∆2e2πikx . (B.5)

and substituting in cos(2x) = 1 − 2sin2(x) we obtain

∇2φ =2

∆2

∑φ(k)sin2(πk∆)e2πikx , (B.6)

which is equal to the right hand side of the Poisson equation. Then we can say that (after

taking a fourier transform)φ(k)

G(k)= ρ(k) (B.7)

175

Page 4: Appendix A List of Symbols · Appendix A List of Symbols The following is a list of symbols used throughout the description of the sticky par-ticle star formation model (see chapter

B. The Green’s Function of the Finite Differenced Laplacian 176

where we have defined the Greens function as

Gj,k,l =

(2

∆2sin2(πk∆)

)−1

(B.8)

Page 5: Appendix A List of Symbols · Appendix A List of Symbols The following is a list of symbols used throughout the description of the sticky par-ticle star formation model (see chapter

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